Icositetrahedron



Triakis octahedron
Tetrakis hexahedron

Deltoidal icositetrahedron
Pentagonal icositetrahedron

In geometry, an icositetrahedron refers to a polyhedron with 24 faces, none of which are regular polyhedra. However, many are composed of regular polygons, such as the triaugmented dodecahedron and the disphenocingulum. Some icositetrahedra are near-spherical, but are not composed of regular polygons. A minimum of 14 vertices is required to form a icositetahedron.

Symmetry

There are many symmetric forms, and the ones with highest symmetry have chiral icosahedral symmetry:
Four Catalan solids, convex:
27 uniform star-polyhedral duals:
Examples with lower symmetry include certain dual polyhedra of Johnson solids, such as the gyroelongated square bicupola and the elongated square gyrobicupola.

Common examples

Common examples include prisms and pyramids, and include certain Johnson solids and Catalan solids.

Icositrigonal pyramids

Icositrigonal pyramids are a type of cone with an icositrigon as a base, with 24 faces, 46 edges, and 24 vertices. Regular icositrigonal pyramids have a regular icositrigon as a base, and its Schläfli symbol is ∨. The surface area and volume with side length and height can be calculated as follows:

Icosidigonal prism

Icosidigonal prisms are a type of cylinder with an icosidigon as a base, with 24 faces, 66 edges, and 44 vertices. Regular icosidigonal prisms have a regular icosidigon as a base, with each face a rectangle. Every vertex borders 2 squares and an icosidigon base. Its vertex configuration is, its Schläfli symbol is × or t, its Coxeter diagram is, and its Conway polyhedron notation is P22. The surface area and volume with side length and height can be calculated as follows:

Hendecagonal antiprism

Hendecagonal antiprisms are antiprisms with a hendecagon as a base, with 24 faces, 44 edges, and 22 vertices. Regular hendecagonal antiprisms have a regular hendecagon as a base, with each face an equilateral triangle. Every vertex borders 2 triangles and a hendecagon base. Its vertex configuration is.

Dodecagonal trapezohedron

Dodecagonal trapezohedra are the tenth member of the trapezohedra family, made of 24 congruent kites arranged radially. Every dodecagonal trapezohedron has 24 faces, 28 edges, and 26 vertices. There are two types of vertices, ones bordering 12 kits and ones bordering 3. Its dual polyhedron is the Hendecagonal antiprism. Its Schläfli symbol is ⨁, its Coxeter diagram is or, and its Conway polyhedron notation is dA12.
Dodecagonal trapezohedra are isohedral figures.

Johnson solids

There are two examples of Johnson solids which are icositetrahedra. They are listed as follows:
NameImageDesignationVerticesEdgesFacesTypes of facesSymmetry groupNet
DisphenocingulumJ9016382420 equilateral triangle,
4 squares		D2d
Triaugmented dodecahedronJ6123452415 equilateral triangles,
9 pentagons		C3v

Catalan Solids

There are 5 types of icositetrahedra with different topologies. The pentagonal icositetetrahedron has two mirror images, so geometrically there are 4 distinct Catalan icositetetrahedra.
NameImageNetDualFacesEdgesVerticesFace ConfigurationPoint Group
Triakis octahedron
Truncated cube243614Isosceles triangle
V3.8.8		Oh
Tetrakis hexahedron
Truncated octahedron243614Isosceles triangle
V4.6.6		Oh
Deltoidal icositetrahedron
Rhombicuboctahedron244826Kite
V3.4.4.4		Oh
Pentagonal icositetrahedron
Snub cube246038irregular pentagon
V3.3.3.3.4		O

Uniform star polyhedra

Some uniform star polyhedra also have 24 faces:
NameImageWythoff symbolVertex figureSymmetry groupFacesEdgesVerticesEuler characteristicDensityFaces by sides
Ditrigonal dodecadodecahedron3 5/3 5
3		Ih246020-16412+12
Dodecadodecahedron5 5/2
5.5/2.5.5/2		Ih246020-16412+12
Truncated great dodecahedron2 5/2 5
10.10.5/2		Ih249060-6312+12
Small stellated truncated dodecahedron2 5 5/3
10/3.10/3.5		Ih249060-6912+12